RUI: Pure and Applied Knot Theory: Skeins, Hyperbolic Volumes, and Biopolymers

纽结理论是对直到连续变形的环的纠缠的数学研究。一个人可以通过将缠绕的线和端点连接起来来创建一个结，如果一个人可以连续地使一个线到另一个线变形，例如通过弯曲、拉伸和通过其他线，但不以任何方式切断或折断线，则两个结是等效的。这个项目既考虑了数学纽结理论的理论问题，也考虑了数学纽结理论的应用。其中一组问题从物理上研究与量子场论有关的纽结不变量。更具体地说，该项目试图了解纽结的量子不变量如何检测纽结的几何属性以及与之相关的3维空间。这项研究的数学方法在数学物理和理论拓扑量子计算中具有潜在的应用前景。另一组问题涉及到在研究打结蛋白质和其他生物聚合物方面的应用，其中一些已知与各种疾病有关。该项目使用纽结理论技术来开发一个模型，该模型可以用来量化局部拓扑复杂性并将其与生物物理过程联系起来。该模型还可用于潜在地设计具有特殊生物物理性质的合成生物聚合物。该项目包括一些适合与本科生合作的研究问题，以及寻求更广泛地提高对数学的兴趣的外联和传播活动。该协会过去曾成功地让本科生参与类似的研究，并将继续建议和鼓励学生继续在数学和相关领域的职业生涯。这项研究分为三部分，两部分试图将量子拓扑学与双曲几何联系起来，一部分将纽结理论应用于分子生物学。一个项目涉及到一个版本的体积猜想，该猜想基于量子拓扑学中的考夫曼括号斜切代数理论，以及它与双曲几何中曲面的TeichMuller空间的关系。第二个项目研究了Kauffman括号代数的一个推广的代数和几何性质，它与带有穿孔的曲面的装饰TeichMuller空间有关。第三个项目涉及与一位生物物理学家合作，研究生物聚合物中通过分子作用力紧密固定的局部纠缠。提出的纽结理论模型将描述这种局部纠缠，使人们能够在实验中量化和测量生物聚合物局部拓扑复杂性的变化。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Knot theory is the mathematical study of entanglement of loops up to continuous deformation. One can create a knot by taking an entangled string and connecting the endpoints, and two knots are equivalent if one can continuously deform one to the other, for example by bending, stretching, and passing strands inside and through others, but without cutting or breaking the string in any way. This project considers both theoretical problems and applications of mathematical knot theory. One group of problems studies a family of invariants of knots related to quantum field theory from physics. More specifically, the project seeks to understand how the quantum invariant of a knot detects geometric properties of the knot and the 3-dimensional spaces that can be associated with it. The mathematical techniques from this research has potential applications to mathematical physics and theoretical topological quantum computing. Another group of problems concerns applications to the study of knotted proteins and other biopolymers, some of which are known to be associated to various diseases. The project uses knot theory techniques to develop a model that can be used to quantify and to relate local topological complexity with biophysical processes. The model can also be used to potentially design synthetic biopolymers with special biophysical properties. The project includes a number of research problems suitable for collaboration with undergraduate students, as well as outreach and dissemination activities that seek to increase interest in mathematics more generally. The PI has successfully involved undergraduate students in similar research in the past and will continue to advise and encourage students to continue careers in mathematics and related areas. The research is split into three parts, two seek to connect quantum topology with hyperbolic geometry and one applies knot theory to molecular biology. One project concerns a version of the Volume Conjecture based the theory of the Kauffman bracket skein algebra from quantum topology and its relationship to the Teichmuller space of a surface from hyperbolic geometry. A second project studies algebraic and geometric properties of a generalization of the Kauffman bracket algebra which is related to the decorated Teichmuller space of a surface with punctures. A third project involves a collaboration with a biophysicist to study local entanglements that are held tightly in place by molecular forces in biopolymers. The proposed knot-theoretic model would give a description of such local entanglements, allowing one to quantify and measure changes in the local topological complexity of biopolymers in experiments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.